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Why Do We Need Counting Methods? When finding a basic probability, what are the two things we need to know? Number of different simple events Number of ways event A can occur P(A) = Sometimes however, it is not practical to construct a list of the outcomes. So finding the total often requires the methods of this section.

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Remember This? A criminal (Mr. Llorens) is found using your social security number and claims that all of the digits were randomly generated. What is the probability of getting your social security number when randomly generating digits? Is the criminal’s (Mr. Llorens) claim that your number was randomly generated likely to be true?

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What We Needed … The Fundamental Counting Rule For a sequence of two (or more) events in which the first can occur m ways and the second can occur n ways, the events together can occur m∙n ways. Apply it! What is the probability of Mr. Llorens randomly generating your social security number? 1/1,000,000,000 =

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Group Review 1 Consider the following question given on a history test: Arrange the following event in chronological order. a) Boston Tea Party b) The invention of the teapot c) The Teapot Dome Scandal d) Ms. P buying her first teapot e) The Civil War Assuming you make a random guess, what is your probability of getting the correct answer?

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Group Review 2 During your upcoming summer vacation, you are planning to visit these six national landmarks: Mrs. A’s birthplace, Mrs. A’s childhood home, Mrs. A’s high school, Mrs. A’s university, Mrs. A’s tennis dominating grounds, Mrs. A’s bird palace. You would like to plan the most efficient route so you can spend as much time at these landmarks as possible. How many different route are there? TRAVESTY NEWS FLASH What if you only have time to visit 4 of these locations? How many possible routes are there now?

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Permutations Rule When Items Are All Different When Some Items Are Identical To Others 1. There are n different items. 2. We select r of the n items (without replacement). 3. We consider rearrangements of the same items to be different. 1. There are n different items, and some are the same. 2. We select all of the n items (without replacement). 3. We consider rearrangements of the same items to be different. IMPORTANT: ORDER MATTERS

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Examples² Horse RacingGardening Woes The 132 nd running of the Kentucky Derby had a field of 20 horses. If a better randomly selects two of those horses to come in 1 st and 2 nd, what is the probability of winning? Lisa is planting 9 flowers in a row. In how many ways can she plant 2 red flowers, 3 yellow flowers, and 4 blue flowers?

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Why is this different? How many ways can 3 class representatives be chosen from a group of 12 students? VS.

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Combinations When Order Doesn’t Matter 1. There are n different items available. 2. We select r of the n items (without replacement). 3. We consider rearrangements of the same items to be the same. (The combination ABC is the same as CBA).

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Solve it! How many ways can 3 class representatives be chosen from a group of 12 students? VS.

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Examples² x 2 Music Taste Movie Choices Find the number of ways to rank to 3 different CDs from a selection of 5 CDs Find the number of ways to rent 5 comedies from a collection of 30 comedies at a Red Box. Careful!

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Multiple Permutations/Combinations Tests Ice Cream! Mrs. Alonso is creating the precalc final exam and has a choice of 10 questions from Chapter Eight, 6 questions from Chapter Thirteen, and 5 questions from Chapter Fifteen. How many distinct tests can be made if she must pick 3 problems from Chapter Eight, 2 problems from Chapter Thirteen, and 2 problems from Chapter Fifteen? Irene’s Ice Cream serves 10 flavors of ice cream, 4 kinds of syrup, and 6 varieties of toppings. How many different Sundaes can you make if each has 2 flavors of ice cream, 2 kinds of syrup, and 3 toppings?